Banach algebras weak* generated by their idempotents

For a closed set E contained in the closed unit interval, we show that the big Lipschitz algebra Lambdagamma (E) (0 < γ < 1) is sequentially weak* generated by its idempotents if and only if it is weak* generated by its idempotents if and only if the little Lipschitz algebra lambda(gamma) (E) is generated by its idempotents, and we describe a class of perfect symmetric sets for which this holds. Moreover, we prove that Lambda(1) (E) is sequentially weak* generated by its idempotents if and only if E is of measure zero. Finally, we show that the quotient algebras A(beta)rootJ(beta)(E)(weak*) of the Beurling algebras need not be weak* generated by their idempotents, when E is of measure zero and beta greater than or equal to 1/2.